Nah, it wouldn't stack.
Maybe if you had two heads, but even then it's not guaranteed.
Stacking isn't the correct word here, but the feat does what it says. It gives you an additional natural attack of your choice and provides a limb to deliver that attack.
So you can indeed gain a second gore attack or even a second bite attack.
This feat is a bit unusual in that it not only provides an attack but a limb, making it 100% clear that the additional attack won't be effected by whatever else you are doing with your limbs/attacks.
Is there any particular rule that says this won't work or stack? The feat says "you grow an additional appendage of your choice" including a gore attack. If multiple claw/slam attacks can exist, why not multiple horn/tusk combinations exist? Let's say I have tusks and a rhino horn, for instance.
I'm looking at this section in the Natural Attacks rule:
"you receive additional attack rolls for multiple limb and body parts capable of making the attack"
and I'm not seeing why this won't work?
Seems like I remembered Mutated Shape wrong, didn't think it actually gave you another appendage. There is a "rule" that you can only gain one attack from one limb/appendage, which is what I thought was the scenario.
So yeah, the reason you can't see why it doesn't work, is because it does work.